A new, approximate substitution principle is presented for a class of steady flows in which both heat transfer and momentum interchange by viscous stresses are significant. The principle, which has important implications for the design and scaling of mixing experiments, can be regarded as an extension of the Munk and Prim substitution principle (for steady isentropic flows) to non-isentropic flows (Munk, M. & Prim, R. C. Proc. natn. Acad. Sci. U.S.A. 33 (5), 137-141 (1947)). The concepts that are developed explain the scaling and distribution of various fluid dynamic properties observed in several different types of flow-mixing experiments. Calculations are done to indicate the expected regimes of applicability of the approximate principle and comparison with experiment is made to show its utility in practical situations.